Finite Group Databasebeta

← #35225090

Subgroups — #35225090

Order 20864.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782217C211yesmaximal, characteristic, complemented
2#23782210C211yesmaximal, characteristic
3#23782062C211yesmaximal, characteristic, complemented
4#23782036C211yesmaximal, characteristic, complemented
5#23782014C211yesmaximal, characteristic, complemented
6#23782010C211yesmaximal, characteristic, complemented
7#23781896C211yesmaximal, characteristic, complemented
8#23781894C211yesmaximal, characteristic, complemented
9#23781892C211yesmaximal, characteristic, complemented
10#23781756C211yesmaximal, characteristic, complemented
11SD16×Dic163C211yesmaximal, characteristic, complemented
12#23781752C211yesmaximal, characteristic, complemented
13#23781750C211yesmaximal, characteristic, complemented
14C2×C8×Dic163C211yesmaximal, characteristic, complemented
15#465×C163C211yesmaximal, characteristic, complemented
16C2×#14579698—21no—
17C2×#14579688C2211yescharacteristic
18C2×#14579687C2211yescharacteristic
19C2×#14579686C2211yescharacteristic
20C22×C163⋊C8C2211yescharacteristic
21#19104734C2211yescharacteristic
22#19104734C2211yescharacteristic
23#19104733C2211yescharacteristic
24#19104731C2211yescharacteristic
25#19104725—21no—
26#19104721—21no—
27#19104714C2211yescharacteristic
28#19104712—21no—
29#19104709C2211yescharacteristic
30#19104707—21no—
31Q8×Dic163C2211yescharacteristic
32#19104700—21no—
33#19104700C2211yescharacteristic
34D4×Dic163C2211yescharacteristic
35#19104697C2211yescharacteristic
36(C22⋊D163)⋊C4—21no—
37#19104691—21no—
38C2×C4×Dic163C2211yescharacteristic
39#19104667C2211yescharacteristic
40#19104666C2211yescharacteristic
41#19104666C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104659C2211yescharacteristic
44#19104658C2211yescharacteristic
45#19104657C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104655C2211yescharacteristic
48C8×Dic163C2211yescharacteristic
49C8×Dic163C2211yescharacteristic
50C8×Dic163C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×SD16×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic
54C22⋊Q8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56#106×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59#14579698—44no—
60C2×Dic326—21no—
61C2×Dic326—21no—
62C2×Dic326C2311yescharacteristic
63C2×D652C2311yescharacteristic
64C2×C22⋊D163—21no—
65C2×C22⋊D163—21no—
66C2×C22⋊D163—21no—
67C22×Dic163—21no—
68C22×Dic163D411yescharacteristic
69C22×Dic163D411yescharacteristic
70C2×C4×D163—21no—
71#14579688—22no—
72#14579687—22no—
73#14579686—22no—
74C2×C163⋊C8—22no—
75C2×C163⋊C8C2311yescharacteristic
76C2×C163⋊C8C2311yescharacteristic
77C4⋊Dic163—21no—
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86C22⋊Dic163—21no—
87C22⋊Dic163—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90C4×Dic163—21no—
91C4×Dic163—21no—
92C4×Dic163D411yescharacteristic
93C4×Dic163D411yescharacteristic
94C4×Dic163C2311yescharacteristic
95C4×Dic163C2311yescharacteristic
96C2×Q8×C163C2311yescharacteristic
97C2×D4×C163—21no—
98C2×D4×C163C2311yescharacteristic
99C22×C4×C163C2311yescharacteristic
100SD16×C163—22no—
101C2×C8×C163—22no—
102C2×C8×C163C2311yescharacteristic
103C2×C8×C163C2311yescharacteristic
104C4⋊C4×C163—21no—
105C4⋊C4×C163C2311yescharacteristic
106C4⋊C4×C163C2311yescharacteristic
107C22⋊C4×C163—21no—
108C22⋊C4×C163—21no—
109C22×D163—21no—
110Dic326—21no—
111Dic326—21no—
112Dic326—42no—
113Dic326—42no—
114D652—21no—
115D652—21no—
116C22⋊D163—42no—
117C22⋊D163—42no—
118C22⋊D163—42no—
119C2×Dic163—22no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—42no—
124C2×Dic163SD1614yescomplemented
125C2×Dic163C2×D411yescharacteristic
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C4×D163—42no—
130C163⋊C8—21no—
131C163⋊C8#4112yescomplemented
132C23×C163—21no—
133C23×C163C2×D411yescharacteristic
134Q8×C163—21no—
135Q8×C163—21no—
136D4×C163—21no—
137D4×C163—21no—
138D4×C163—42no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2411yescharacteristic
146C8×C163—21no—
147C8×C163#4112yescomplemented
148C22×C163—21no—
149C22×C163—21no—
150C22×C163—41no—
151C22×C163—41no—
152C22×C163—41no—
153C22×C163C2×SD1612yes—
154C22×C163C22×D411yescharacteristic
155C2×D163—41no—
156C2×D163—41no—
157C2×D163—41no—
158C4×C163—21no—
159C4×C163—41no—
160C4×C163—41no—
161C4×C163—41no—
162C4×C163C2×#4111yescharacteristic
163C4×C163C2×#4111yescharacteristic
164Dic163—21no—
165Dic163—41no—
166Dic163—41no—
167Dic163—41no—
168Dic163C2×SD1612yes—
169Dic163C2×SD1612yes—
170Dic163#13512yes—
171C2×C163—22no—
172C2×C163—81no—
173C2×C163#53211yescharacteristic
174C2×C163C22×SD1611yescharacteristic
175C2×C163C2×#13511yescharacteristic
176D163—81no—
177C163#316711yescharacteristic, complemented, Sylow
178#3167—1631nomaximal, Sylow
179#533—1631no—
180#532—1631no—
181#498—1631no—
182#492—1631no—
183#487—1631no—
184#486—1631no—
185#465—1631no—
186#465—1631no—
187#465—1631no—
188#465D16311yescharacteristic, complemented
189C4×SD16—1631no—
190C4×SD16—1631no—
191C4×SD16—1631no—
192C4×SD16—1631no—
193C2×C4×C8—1631no—
194C2×#41—3261no—
195C2×SD16—1631no—
196C2×SD16—1631no—
197C2×SD16—1631no—
198C2×SD16C2×D16311yescharacteristic
199C22×C8—1631no—
200C22×C8C2×D16311yescharacteristic
201C4⋊Q8—1631no—
202C4⋊Q8—1631no—
203C4⋊Q8—1631no—
204#127—1631no—
205#125—3261no—
206#124—3261no—
207C22⋊Q8—3261no—
208C22⋊Q8—1631no—
209C22⋊Q8C2×D16311yescharacteristic
210#121—3261no—
211#121—1631no—
212#121C2×D16311yescharacteristic
213C4×Q8—3261no—
214C4×Q8—1631no—
215C4×Q8—1631no—
216C4×D4—3261no—
217C4×D4—1631no—
218C4×D4—1631no—
219#117—3261no—
220C2×C42—1631no—
221#106—1631no—
222#106—1631no—
223#106—1631no—
224#106C2×D16311yescharacteristic
225#103—1631no—
226#103—1631no—
227#103—1631no—
228#103C2×D16311yescharacteristic
229#102—1631no—
230#102—1631no—
231#102—1631no—
232#102C2×D16311yescharacteristic
233C4×C8—1631no—
234C4×C8—1631no—
235C4×C8—1631no—
236C4×C8—1631no—
237#41—6524no—
238C2×Q8—3261no—
239C2×Q8—3261no—
240C2×Q8—1631no—
241C2×Q8C22×D16311yescharacteristic
242C2×D4—21no—
243C2×D4—3261no—
244C2×D4—3261no—
245C2×D4—3261no—
246C2×D4—1631no—
247C2×D4C22×D16311yescharacteristic
248C22×C4—3261no—
249C22×C4—3261no—
250C22×C4—1631no—
251C22×C4—1631no—
252C22×C4C22×D16311yescharacteristic
253SD16—22no—
254SD16—3262no—
255SD16—3262no—
256SD16—3262no—
257C2×C8—22no—
258C2×C8—3262no—
259C2×C8—1631no—
260C2×C8—1631no—
261C2×C8C22×D16311yescharacteristic
262C2×C8C22×D16311yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—3261no—
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—3261no—
271C4⋊C4—1631no—
272C4⋊C4—1631no—
273C4⋊C4C22×D16311yescharacteristic
274C4⋊C4C22×D16311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—3261no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C42—3261no—
282C42—3261no—
283C42—1631no—
284C42—1631no—
285C42—1631no—
286C42—1631no—
287C23—21no—
288C23—3261no—
289C23D4×D16311yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—3261no—
293Q8—3261no—
294Q8—6522no—
295Q8—6522no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—3261no—
300D4—3261no—
301D4—6522no—
302D4—6522no—
303D4—6522no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—3262no—
310C2×C4—3261no—
311C2×C4—3261no—
312C2×C4—3261no—
313C2×C4—6522no—
314C2×C4—6522no—
315C2×C4—1631no—
316C2×C4—1631no—
317C2×C4—1631no—
318C2×C4—1631no—
319C2×C4—1634no—
320C2×C4D4×D16311yescharacteristic
321C2×C4C23×D16311yescharacteristic
322C8—21no—
323C8—3261no—
324C8—1632no—
325C8#1457969812yescomplemented
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—6521no—
332C22—6521no—
333C22—6521no—
334C22SD16×D16312yes—
335C22C2×D4×D16311yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—3261no—
341C4—6521no—
342C4—6521no—
343C4—6521no—
344C4—1632no—
345C4—1632no—
346C4—1632no—
347C4#41×D16311yescharacteristic
348C4C2×#1457969811yescharacteristic
349C2—22no—
350C2—81no—
351C2—13041no—
352C2#2378221011yescharacteristic
353C2C2×SD16×D16311yescharacteristic
354C2#135×D16311yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#465×C163C2
socleC22×C163C22×D4
prime radical#23782210C2
hypercenter#465D163
hypocenterC163#3167
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #23782217C2semidirect
3. #23782062C2semidirect
4. #23782036C2semidirect
5. #23782014C2semidirect
6. #23782010C2semidirect
7. #23781896C2semidirect
8. #23781894C2semidirect
9. #23781892C2semidirect
10. #23781756C2semidirect
11. SD16×Dic163C2semidirect
12. #23781752C2semidirect
13. #23781750C2semidirect
14. C2×C8×Dic163C2semidirect
15. #465×C163C2semidirect
124. C2×Dic163SD16semidirect
131. C163⋊C8#41semidirect
147. C8×C163#41semidirect
177. C163#3167semidirect
188. #465D163semidirect
325. C8#14579698semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237822172
#237822102
#237820622
#237820362
#237820142
#237820102
#237818962
#237818942
#237818922
#237817562
SD16×Dic1632
#237817522
#237817502
C2×C8×Dic1632
#465×C1632
#3167163